Revisiting Bresse-Timoshenko theory for beams

In this study, a variational derivation of the simpler and more consistent version of Bresse-Timoshenko
beams equations, taking into account both shear deformation and rotary inertia in vibrating beams, is
presented. Whereas Timoshenko gets his beam equations in terms of the equilibrium, the governing
equations and the boundary conditions are here derived using the Hamilton’s principle. First, a list of
the different energy contributions is established, including the shear effect and the rotary inertia.
Second, the Hamilton’s principle is applied demanding the stationary of an appropriate functional,
leading to two different equations of motion. The resolution of these equations provides the governing
differential equation. It turns out that an additional term appears. The derived equations are intended for
dynamic stability applications. Specifically, the parametric vibrations will be studied when the axial force
varies periodically. This problem has important aerospace applications.